m Moduli of Smoothness Related to the Laplace-Operator

We introduce and study a series of new moduli of smoothness in the multivariate case in L-p-spaces of periodic functions. The main focus lies on the case 0 < p < 1. We prove a direct Jackson-type estimate and provide necessary and sufficient conditions with respect to the dimension d and to integrability p for the equivalence of these moduli and polynomial K-functionals related to the Laplace-operator. As a consequence we obtain an inverse Bernstein-type estimate. Moreover, we are able to characterize the approximation error in case of approximation by families of linear polynomial operators which are generated by Bochner-Riesz kernels in terms of the introduced moduli.